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Section Duration Data Introduction Sometimes we have data on length of time of a particular event or ‘spells’ Time until death Time on unemployment Time to complete a PhD

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Section

Duration Data


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Introduction

  • Sometimes we have data on length of time of a particular event or ‘spells’

    • Time until death

    • Time on unemployment

    • Time to complete a PhD

  • Techniques we will discuss were originally used to examine lifespan of objects like light bulbs or machines. These models are often referred to as “time to failure”


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Notation

  • T is a random variable that indicates duration (time til death, find a new job, etc)

  • t is the realization of that variable

  • f(t) is a PDF that describes the process that determines the time to failure

  • CDF is F(t) represents the probability an event will happen by time t


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  • F(t) represents the probability that the event happens by ‘t’.

  • What is the probability a person will die on or before the 65th birthday?


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  • Survivor function, what is the chance you live past (t)

  • S(t) = 1 – F(t)

  • If 10% of a cohort dies by their 65th birthday, 90% will die sometime after their 65th birthday


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  • Hazard function, h(t)

  • What is the probability the spell will end at time t, given that it has already lasted t

  • What is the chance you find a new job in month 12 given that you’ve been unemployed for 12 months already


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  • PDF, CDF (Failure function), survivor function and hazard function are all related

  • λ(t) = f(t)/S(t) = f(t)/(1-F(t))

  • We focus on the ‘hazard’ rate because its relationship to time indicates ‘duration dependence’


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  • Example: suppose the longer someone is out of work, the lower the chance they will exit unemployment – ‘damaged goods’

  • This is an example of duration dependence, the probability of exiting a state of the world is a function of the length


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  • Mathematically

    • d λ(t) /dt = 0 then there is no duration dep.

    • d λ(t) /dt > 0 there is + duration dependence

      the probability the spell will end

      increases with time

    • d λ(t) /dt < 0 there is – duration dependence

      the probability the spell will end

      decreases over time


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  • Your choice, is to pick values for f(t) that have +, - or no duration dependence


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Different Functional Forms

  • Exponential

    • λ(t)= λ

    • Hazard is the same over time, a ‘memory less’ process

  • Weibull

    • F(t) = 1 – exp(-γtα) where α,γ > 0

    • λ(t) = αγtα-1

    • if α>1, increasing hazard

    • if α<1, decreasing hazard

    • if α=1, exponential


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  • Others: Lognormal, log-logistic, Gompertz


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NHIS Multiple Cause of Death

  • NHIS

    • annual survey of 60K households

    • Data on individuals

    • Self-reported healthm DR visits, lost workdays, etc.

  • MCOD

    • Linked NHIS respondents from 1986-1994 to National Death Index through Dec 31, 1995

    • Identified whether respondent died and of what cause


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  • Our sample

    • Males, 50-70, who were married at the time of the survey

    • 1987-1989 surveys

    • Give everyone 5 years (60 months) of followup


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Key Variables

  • max_mths maximum months in the survey.

  • Diedin5 respondent died during the 5 years of followup

  • Note if diedn5=0, the max_mths=60. Diedin5 identifies whether the data is censored or not.


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Identifying Duration Data in STATA

  • Need to identify which is the duration data

    stset length, failure(failvar)

    • Length=duration variable

    • Failvar=1 when durations end in failure, =0 for censored values

  • If all data is uncensored, omit failure(failvar)


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    • In our case

    • Stset max_mths, failure(diedin5)


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    Getting Kaplan-Meier Curves

    • Tabular presentation of results

      sts list

    • Graphical presentation

      sts graph

    • Results by subgroup

      sts graph, by(income)


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